INMO 1993

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$ . Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$ . Show that the points $H,P,O$ are collinear.

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$ . Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$ .

If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$ , show that $$ab +bc +cd \leq \dfrac{1}{4}.$$

Let $ABC$ be a triangle in a plane $\pi$ . Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$ , $BCP$ , $CAP$ have the same radii.

Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.

Let $ABC$ be a triangle right-angled at $A$ and $S$ be its circumcircle. Let $S_1$ be the circle touching the lines $AB$ and $AC$ , and the circle $S$ internally. Further, let $S_2$ be the circle touching the lines $AB$ and $AC$ and the circle $S$ externally. If $r_1, r_2$ be the radii of $S_1, S_2$ prove that $r_1 \cdot r_2 = 4 A[ABC]$ .